How we describe the directionality of an effect affects how we think about it. Andrew Gelman complains that the recent paper by Dalton Conley and Emily Rauscher does this. It’s called, “The Effect of Daughters on Partisanship and Social Attitudes Toward Women.” And the news headlines were things like, “Does Having Daughters Make You More Republican?” Ross Douthat called it “The Daughter Theory.”

But of course the finding could just as well be described as the effect of sons on making people more liberal.

In this case it’s a great example of boys being the norm and girls being difference. But there are plenty of examples of when we describe an effect as if its opposite doesn’t exist. Here are three:

The marriage premium. This usually refers to married men earning more than single men. But it is just as much a penalty for being single as it is a reward for being married. In my own work on this I described three possible mechanisms: positive selection into marriage (higher earners marry), productivity-enhancing effects of marriage (wives make men better workers), and discrimination (bosses prefer married men). But all of these could have been expressed in the reverse direction. Lots of “marriage is good” arguments should be turned around to ask, “How could we punish single people less”?

The gender gap. President Obama has frequently implied that reducing the gender gap in pay will be good for “middle class families.” Under “Protecting the Middle Class News,” the White House writes that the gender gap “means less for families’ everyday needs, less for investments in our children’s futures, and, when added up over a lifetime of work, substantially less for retirement.” Of course, it also means more for families with employed men. I hate to be a buzzkill on this, but there is no reason to think that reducing gender discrimination just means paying women more. How do we know women are underpaid, instead of men being overpaid?

Returns to education. This one is tricky, because there is a return on investment from education, so it’s reasonable to talk about the effect in that direction: you spend money on education, you get a benefit. But the society that rewards education also penalizes lack of education relatively speaking (unless everyone is equally educated). Nothing against educated people, but to reduce inequality it would be good to reduce the returns to education. For example, raising the minimum wage, or providing government jobs to low-skilled workers, would reduce returns to education (if that is operationalized as the difference between college and non-college wages.

Easy to illustrate if you think of the regression of earnings on education as a structural equation (like we all did back during the Ford administration). I do this in my stat classes to make precisely the point you raise (and it relates to the exchange with Carter B from earlier in the week). Y is earnings (or log earnings), X is education, and e is the impact of unmeasured factors, and Y = b*X + e (everything is centered so there’s no intercept). Variance of Y (inequaltiy in earnings) = b**2*Var(X) + Var(e). If this really is a structural equation (in the Duncan sense) then the variance in earnings COMES FROM three sources. It increases when the impact of earnings (b) increases, it increases when the inequality of education [Var(X)] increases, and it increases when the variation in everything else [conditional inequality, Var(e)] increases. AND it declines when any of those quantities decreases. Even more illuminating when X1 is education and X2 is social origins, so Var (Y) = b1**2*Var(X1) + b2**Var(X2) + 2*b1*b2*Cov(X1,X2) + Var(e) .Every one of those terms is substantively interesting with a policy implication. Introduce the possibility of covariation between e and either X1 or X2, and you can start talking about specificaion error. Yes, this is old fasioned in the era of causality as counterfactual, but almost all of us still use the language of causality (and of omitted variable bias and specificaion error) even if we insist we’re not structural equationalists). Barack, meet Otis Dudley Duncan (via Sewell Wright and Art Goldberger).

Freakonomics author Stephen Dubner wrote, “If you’re a child who’s adopted into a high-education family — that is where the parents both went to college — you are about 16 percentage points more likely to go to college than a kid who gets adopted into a low-education family. So that sounds pretty good, OK?”

But, if you flip it around, the study is saying that if you’re a child who’s adopted into a low-education family you are about 16 percentage points less likely to go to college than a kid who gets adopted into a high-education family.

Got that? Just for being adopted into the wrong kind of family, your chance of going to college goes down by 16%! That doesn’t sound so good now, does it? Which makes you wonder why Dubner was so sure that the +16% effect (which of course is the same thing, just looked at from the other direction) sounded so good.

I’m not attributing any malice to Dubner here. I just think this happens all the time, that people look at a comparison without thinking of it in the opposite direction.

Some of the issues talked about are incorrect; y=f(X) may be a valid problem; but x = f^(-1)y may be invalid or singular. At the worst, it can lead to the famous confusion of inverse. That is, given two events A and B, the probability of A happening given that B has happened is assumed to be about the same as the probability of B given A. More formally, P(A|B) is assumed to be approximately equal to P(B|A). This is always incorrect.

Talk about a counterexample:

If I work hard in my classes, I will get an A; however, if the rest of the class does less than average, and get Cs, then nobody should argue that the less-hardworking person should be punished less, and we should reduce the impact on others by giving everyone As (which is happening a lot now).

Almost all of your 3 examples, is lessening the impact of f(B/A); it goes without saying that this will lead to less possible outcome of A. Yes, I am saying that if zero the marriage premium, very few people will marry, although you had the “good” intention of helping out the unmarried (I so understand that the only good marriage now is gay marriage).